[Paper Review] A field-level emulator for modified gravity

Daniela Saadeh, K. Koyama+1 more|arXiv (Cornell University)|Jun 5, 2024

Geophysics and Gravity Measurements4 citations

TL;DR

This paper presents a field-level neural network emulator that transforms ΛCDM N-body simulation outputs into equivalent simulations under f(R) modified gravity, using the gravity strength |f_R0| as a style condition. Trained on fixed cosmologies, it achieves 1% accuracy in the nonlinear matter power spectrum up to k ~ 1 h/Mpc and shows strong extrapolation performance to unseen cosmological parameters.

ABSTRACT

Stage IV surveys like LSST and Euclid present a unique opportunity to shed light on the nature of dark energy. However, their full constraining power cannot be unlocked unless accurate predictions are available at all observable scales. Currently, only the linear regime is well understood in models beyond $Λ$CDM: on the nonlinear scales, expensive numerical simulations become necessary, whose direct use is impractical in the analyses of large datasets. Recently, machine learning techniques have shown the potential to break this impasse: by training emulators, we can predict complex data fields in a fraction of the time it takes to produce them. In this work, we present a field-level emulator capable of turning a $Λ$CDM N-body simulation into one evolved under $f(R)$ gravity. To achieve this, we build on the map2map neural network, using the strength of modified gravity $|f_{R_0}|$ as style parameter. We find that our emulator correctly estimates the changes it needs to apply to the positions and velocities of the input N-body particles to produce the target simulation. We test the performance of our network against several summary statistics, finding $1\%$ agreement in the power spectrum up to $k \sim 1$ $h/$Mpc, and $1.5\%$ agreement against the independent boost emulator eMantis. Although the algorithm is trained on fixed cosmological parameters, we find it can extrapolate to models it was not trained on. Coupled with available field-level emulators and simulation suites for $Λ$CDM, our algorithm can be used to constrain modified gravity in the large-scale structure using full information available at the field level.

Motivation & Objective

• To overcome the computational cost of running full N-body simulations for modified gravity models like f(R) across diverse cosmological parameters.
• To enable field-level inference in large-scale structure cosmology by preserving phase and morphological information lost in summary statistics.
• To develop a differentiable, generalizable emulator that can extrapolate beyond its training cosmologies to accelerate cosmological model testing.
• To integrate with existing ΛCDM field-level emulators (e.g., map2map) for end-to-end, high-fidelity testing of modified gravity in upcoming Stage IV surveys.
• To support advanced inference frameworks like BORG by providing accurate, fast predictions of matter and velocity fields under modified gravity.

Proposed method

• Adapts the map2map neural network architecture to serve as a field-level emulator that maps ΛCDM particle positions and velocities to their f(R) gravity counterparts.
• Uses the absolute value of the f(R) parameter |f_R0| as a style embedding to condition the network’s transformation of the input N-body field.
• Trains the model on a set of ΛCDM simulations with varying |f_R0| values but fixed cosmological parameters (ΩM, σ8, h, etc.).
• Employs a loss function that minimizes differences in key cosmological statistics: power spectrum, bispectrum, density and momentum fields, and redshift-space distortions.
• Validates performance using independent simulations from the ECOSMOG AMR code and compares against the eMantis emulator for boost factor accuracy.
• Evaluates generalization via zero-shot extrapolation to cosmologies not in the training set, including extreme values of ΩM, σ8, and h.

Figure 1: Comparison of the emulator’s output against the target snapshot. In this example, $|f_{R_{0}}|$ $=10^{-5}$ . The top, middle, and bottom rows show the projected density field reconstructed from the snapshots, along three perpendicular directions. The left and middle columns show the emulat

Experimental results

Research questions

• RQ1 Can a deep learning emulator accurately transform ΛCDM N-body simulations into f(R) gravity simulations at the field level, preserving both amplitude and phase information?
• RQ2 To what extent can such an emulator generalize to cosmological parameters outside its training distribution, particularly in ΩM, σ8, and h?
• RQ3 How accurately does the emulator reproduce key observables like the nonlinear matter power spectrum, bispectrum, and redshift-space distortions?
• RQ4 Can the emulator achieve sub-percent-level accuracy in summary statistics across scales relevant to Stage IV surveys (k < 1 h/Mpc)?
• RQ5 How does the emulator’s performance compare to established emulators like eMantis, especially in predicting the growth rate boost factor?

Key findings

• The emulator achieves 1% agreement with target simulations in the nonlinear matter power spectrum up to k ~ 1 h/Mpc.
• It matches the performance of the independent eMantis emulator to within 1.5% in the boost factor across the same scale range.
• The emulator reproduces the bispectrum with ~1% accuracy, demonstrating preservation of phase and morphological structure.
• Redshift-space distortion monopole and quadrupole predictions agree with target simulations within 4% at scales k < 0.3 h/Mpc.
• Stochasticity in the emulated density and momentum fields is below 0.4% at k < 1 h/Mpc and k < 0.3 h/Mpc, respectively.
• The model generalizes well to unseen cosmologies, achieving ~3% accuracy in the boost factor and ~1.5% in the bispectrum across diverse ΩM, σ8, and h values.

Figure 2: Relative error on the boost factor ${P(k)}^{\mathrm{(MG)}}/{P(k)}^{\mathrm{(\Lambda CDM)}}$ as computed from the 2000 FML simulations in the traning, validation and test datasets, and using the screening efficiencies listed in Table 1 . For this comparison, we use eMantis as the ground tru